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In the racetrack shown above, regions I and III are semi-circular with radius r. If region II is rectangular and its length is twice its width, what is the perimeter of the track in terms of r? – OGQR 2020 Question #94 with Solution

OGQR 2020: Question No. 94

In the racetrack shown above, regions I and III are semi-circular with radius r. If region II is rectangular and its length is twice its width, what is the perimeter of the track in terms of r?

Source OGQR 2020
Type Problem Solving
Topic Geometry
Sub-Topic Circle / Rectangle
Difficulty Medium

Solution

Given

In this question, we are given

  • The diagram of a racetrack consists of two semi-circular regions, with radius r, and one rectangular region.
  • The length of the rectangular region is twice its width.

To Find

We need to determine

  • The perimeter of the track, expressed in terms of r.

Approach & Working

The perimeter of each semi-circular region = πr

  • Therefore, the perimeter of both semi-circular regions together = 2πr

The rectangular region has length twice its width.

  • Width of the rectangular region = diameter of the semi-circular region = 2r
  • Hence, length of the rectangular region = 2 * 2r = 4r

Therefore, the perimeter of the whole region = 2πr + 4r + 4r = 2πr + 8r = 2r (π + 4)

Hence, the correct answer is option B.

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